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''polynomial distribution''
 
''polynomial distribution''
  
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[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
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The joint distribution of random variables $  X _ {1} \dots X _ {k} $
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The joint distribution of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653301.png" /> that is defined for any set of non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653302.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653305.png" />, by the formula
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that is defined for any set of non-negative integers $  n _ {1} \dots n _ {k} $
 
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satisfying the condition $  n _ {1} + \dots + n _ {k} = n $,  
 
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$  n _ {j} = 0 \dots n $,  
 
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$  j = 1 \dots k $,  
 
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by the formula
 
  
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$$ \tag{* }
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{\mathsf P} \{ X _ {1} = n _ {1} \dots X _ {k} = n _ {k} \}  = \
 
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n! over {n _ {1} ! \dots n _ {k} ! } p _ {1} ^ {n _ {1} } \dots p _ {k} ^ {n _ {k} } ,
 
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$$
 
  
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where $  n, p _ {1} \dots p _ {k} $(
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653307.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653309.png" />) are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533011.png" /> (this distribution is in essence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533012.png" />-dimensional, since it is degenerate in the Euclidean space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533013.png" /> dimensions). A multinomial distribution is a natural generalization of a [[Binomial distribution|binomial distribution]] and coincides with the latter for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533014.png" />. The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533015.png" />. The multinomial distribution appears in the following probability scheme. Each of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533016.png" /> is the number of occurrences of one of the mutually exclusive events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533018.png" />, in repeated independent trials. If in each trial the probability of event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533021.png" />, then the probability (*) is equal to the probability that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533022.png" /> trials the events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533023.png" /> will appear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533024.png" /> times, respectively. Each of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533025.png" /> has a binomial distribution with mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533026.png" /> and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533027.png" />.
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$  p _ {j} \geq  0 $,  
 
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$  \sum p _ {j} = 1 $)  
 
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are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector $  ( X _ {1} \dots X _ {k} ) $
 
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with $  X _ {1} + \dots + X _ {k} = n $(
 
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this distribution is in essence $  ( k- 1) $-
 
−
dimensional, since it is degenerate in the Euclidean space of $  k $
 
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dimensions). A multinomial distribution is a natural generalization of a [[Binomial distribution|binomial distribution]] and coincides with the latter for $  k = 2 $.  
 
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The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial $  ( p _ {1} + \dots + p _ {k} )  ^ {n} $.  
 
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The multinomial distribution appears in the following probability scheme. Each of the random variables $  X _ {i} $
 
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is the number of occurrences of one of the mutually exclusive events $  A _ {j} $,  
 
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$  j = 1 \dots k $,  
 
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in repeated independent trials. If in each trial the probability of event $  A _ {j} $
 
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is $  p _ {j} $,  
 
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$  j = 1 \dots k $,  
 
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then the probability (*) is equal to the probability that in $  n $
 
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trials the events $  A _ {1} \dots A _ {k} $
 
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will appear $  n _ {1} \dots n _ {k} $
 
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times, respectively. Each of the random variables $  X _ {j} $
 
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has a binomial distribution with mathematical expectation $  np _ {j} $
 
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and variance $  np _ {j} ( 1- p _ {j} ) $.
 
  
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The random vector $  ( X _ {1} \dots X _ {k} ) $
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The random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533028.png" /> has mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533029.png" /> and covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533030.png" />, where
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has mathematical expectation $  ( np _ {1} \dots np _ {k} ) $
 
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and covariance matrix $  B = \| b _ {ij} \| $,  
 
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where
 
  
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533031.png" /></td> </tr></table>
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b _ {ij}  = \left \{
 
  
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(the rank of the matrix $  B $
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(the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533032.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533033.png" /> because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533034.png" />). The characteristic function of a multinomial distribution is
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is $  k- 1 $
 
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because $  \sum _ {i=} 1  ^ {k} n _ {i} = n $).  
 
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The characteristic function of a multinomial distribution is
 
  
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533035.png" /></td> </tr></table>
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f( t _ {1} \dots t _ {k} )  = \left ( p _ {1} e ^ {it _ {1} } + \dots + p _ {k} e ^ {it _ {k} } \right )  ^ {n} .
 
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$$
 
  
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For $  n \rightarrow \infty $,  
+
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533036.png" />, the distribution of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533037.png" /> with normalized components
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the distribution of the vector $  ( Y _ {1} \dots Y _ {k} ) $
 
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with normalized components
 
  
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$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533038.png" /></td> </tr></table>
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Y _ {i}  = \
 
−
 
 
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\frac{X _ {i} - np _ {i} }{\sqrt {np _ {i} ( 1- p _ {i} ) } }
 
−
 
 
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$$
 
  
 
tends to a certain multivariate [[Normal distribution|normal distribution]], while the distribution of the sum
 
tends to a certain multivariate [[Normal distribution|normal distribution]], while the distribution of the sum
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533039.png" /></td> </tr></table>
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\sum _ { i= } 1 ^ { k }  ( 1 - p _ {i} ) Y _ {i}  ^ {2}
 
−
$$
 
  
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(which is used in mathematical statistics to construct the [[Chi-squared distribution| "chi-squared" test]]) tends to the [[Chi-squared test| "chi-squared" distribution]] with $  k- 1 $
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(which is used in mathematical statistics to construct the [[Chi-squared distribution| "chi-squared" test]]) tends to the [[Chi-squared test| "chi-squared" distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533040.png" /> degrees of freedom.
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degrees of freedom.
 
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
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====References====
 
====References====

Revision as of 14:32, 7 June 2020

polynomial distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

The joint distribution of random variables that is defined for any set of non-negative integers satisfying the condition , , , by the formula

(*)

where (, ) are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector with (this distribution is in essence -dimensional, since it is degenerate in the Euclidean space of dimensions). A multinomial distribution is a natural generalization of a binomial distribution and coincides with the latter for . The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial . The multinomial distribution appears in the following probability scheme. Each of the random variables is the number of occurrences of one of the mutually exclusive events , , in repeated independent trials. If in each trial the probability of event is , , then the probability (*) is equal to the probability that in trials the events will appear times, respectively. Each of the random variables has a binomial distribution with mathematical expectation and variance .

The random vector has mathematical expectation and covariance matrix , where

(the rank of the matrix is because ). The characteristic function of a multinomial distribution is

For , the distribution of the vector with normalized components

tends to a certain multivariate normal distribution, while the distribution of the sum

(which is used in mathematical statistics to construct the "chi-squared" test) tends to the "chi-squared" distribution with degrees of freedom.

References

[C] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014

Comments

References

[JK] N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969) MR0268996 Zbl 0292.62009
How to Cite This Entry:
Multinomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multinomial_distribution&oldid=49316
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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